In this work, we consider the classic elliptic PDE`s coefficient inverse problem in which repeated forward and adjoint solves incur substantial computational cost during iterative inversion. To address this challenge, we propose a hybrid reduced-full order modeling (ROM-FOM) framework that combines proper orthogonal decomposition (POD) with selective full-order corrections. The proposed approach employs a POD-based reduced-order model to approximate both state and adjoint solutions within a gradient-based framework. To enhance robustness, a warm-start full-order solve is introduced in the early iterations, stabilizing updates and limiting error accumulation. In the theoretical analysis, we investigate the consistency of the ROM formulation and how errors propagate through the state, adjoint, and gradient computations. We also demonstrate numerical results to show how the proposed method achieves a balance between efficiency and accuracy. This framework provides a practical and extensible approach for large-scale PDE-constrained inverse problems and can be extended to time-dependent settings.